If YYY is a uniform space (for example, if YYY is a metric space), then this is the topology of uniform convergence on compact sets. That is, a sequence(fn)subscriptfn\left(f_{n}\right)converges to fff in the compact-open topology if and only if for every compact subspace KKK of X,XX,(fn)subscriptfn\left(f_{n}\right) converges to fff uniformly on KKK. If in additionXXX is a compact space, then this is the topology of uniform convergence.